Reflection spectroscopy is a method for investigating layer systems that is widespread and has been used for some time. In the manufacture of semiconductor chips, for example, semiconductor wafers are coated with thin layers. For reasons of quality control, it is then desirable to determine the thickness of those layers without damaging the semiconductor wafer. Such investigations can be performed, for example, using reflection spectroscopy. The principle of this method is very simple: a specimen having multiple layers is irradiated with light of a specified wavelength. If the layers are transparent, the light penetrates into the layers and is partially reflected in the transition regions between two layers (also including the transition between the topmost layer and the surrounding atmosphere). Superposition of the incident and reflected light results in interferences, thus influencing the intensity of the reflected light. The ratio between the intensities of the incident and reflected light determines the so-called absolute reflectance; both intensities therefore need to be measured. If the wavelength is then varied continuously within a specified range, the reflection spectrum is then obtained; this exhibits, as a function of wavelength, maxima and minima that are caused by the interferences. The locations of these extremes depend on the material properties of the specimen which determine its optical behavior, including especially the thicknesses of the individual layers.
It is possible in principle to deduce the layer thicknesses from the measured reflection spectrum. In an ideal model, the limits in terms of the thickness of the layers and their quantity are extremely broad. The underlying formulae can be derived from Fresnel diffraction theory, as described in detail in the article by P. S. Hauge, “Polycrystalline silicon film thickness measurement from analysis of visible reflectance spectra” in J. Opt. Soc. Am., Vol. 69 (8), 1979, pp. 1143–1152. As is evident from the book by O. Stenzel, “Das Dünnschichtspektrum” (The thin-layer spectrum), Akademieverlag 1996, pp. 77–80, however, determining the optical constants and layer thicknesses by back-calculation is in reality very difficult and laborious, since the number of unknowns is very large.
Regardless of the model and approximations used, a determination of layer thicknesses generally proceeds in the following manner for a system having multiple layers: For each layer, a reflection spectrum is modeled in a specified wavelength range for layer thickness values that lie, spaced apart by a specified increment, between a minimum and a maximum value. The thicknesses of the other layers are each kept constant. Using this first coarse search method, it is possible to determine the ranges within which the layer thicknesses of the individual layers can lie, by (in simplified terms) comparing the numbers of extremes in the measured and in the modeled reflection spectrum. If those numbers deviate from one another by more than a specified difference, those thickness values are discarded. A more refined search method can then be applied, within the thickness ranges that have been identified, in order to determine the actual thickness values.
A coarse search method for delimiting layer thickness ranges is disclosed, for example, in U.S. Pat. No. 5,493,401, which also represents the closest existing art. In the model taken as the basis therein, the number m of extremes between two wavelengths λs and λe, which respectively mark the start and end points of the measured spectrum, are determined using the equation                     m        =                  4          ⁢                                    ∑                              i                =                1                            L                        ⁢                                          (                                                      〈                                          n                      i                                        〉                                    ×                                      d                    i                                                  )                            ×                                                                                          λ                      s                                        -                                          λ                      e                                                                                                  λ                      s                                        ⁢                                          λ                      e                                                                      .                                                                        (        1        )            
The sum i extends over all layers, <n1> is the average refractive index for that layer within the given wavelength range, and di is the layer thickness selected for layer i. The layer thickness is varied for each layer between 0 and an upper limit value di,max, the upper limit value being specified using the formula                                           d                          i              ,              max                                =                                    1              4                        ⁢                                          m                +                1                +                γ                                            〈                                  n                  i                                〉                                      ×                                                            λ                  s                                -                                  λ                  e                                                                              λ                  s                                ⁢                                  λ                  e                                                                    ,                            (        2        )            where ε designates a safety factor and is indicated as 0.2. Within this range, the layer thicknesses are scanned at constant increments Δdi, the increments for each layer being defined as a function of the average refractive index of that layer and the limit wavelengths, using the equation                               Δ          ⁢                                          ⁢                      d            i                          =                              0.1                          4              ⁢                              〈                                  n                  i                                〉                            ×                                                                    λ                    e                                    -                                      λ                    s                                                                                        λ                    s                                    ⁢                                      λ                    e                                                                                .                                    (        3        )            
This method has several disadvantages, however. For example, it is assumed that the measured spectrum can be evaluated within the entire wavelength range. No account is taken of the fact that this may be impossible in some circumstances due to noise and/or an insufficient signal. The limit values for the layer thicknesses are also estimated only very roughly; especially when 0 is set as the lower limit, this results in an unnecessary number of operations that ultimately are discarded. Far more serious, however, is the disadvantage that absorption is not taken into consideration in this method. For this reason, the method can function only with transparent layers; this unnecessarily limits the class of materials that can be investigated, and even in that case the results are reliable only for non dispersive or very weakly dispersive materials, since one and the same refractive index, namely the average <ni>, is used for all wavelengths. This method furthermore fails when fluctuations occur in the reflection spectrum, which can certainly occur with multiple-layer structures and if the angle of incidence of the light onto the specimen is not perpendicular.